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Properties

Label 2-182-7.2-c1-0-0
Degree $2$
Conductor $182$
Sign $0.156 - 0.987i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1.18 − 2.04i)3-s + (−0.499 − 0.866i)4-s + (−1.97 + 3.41i)5-s + 2.36·6-s + (2.58 + 0.578i)7-s + 0.999·8-s + (−1.28 + 2.23i)9-s + (−1.97 − 3.41i)10-s + (2.76 + 4.78i)11-s + (−1.18 + 2.04i)12-s − 13-s + (−1.79 + 1.94i)14-s + 9.30·15-s + (−0.5 + 0.866i)16-s + (1.97 + 3.41i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.681 − 1.18i)3-s + (−0.249 − 0.433i)4-s + (−0.881 + 1.52i)5-s + 0.964·6-s + (0.975 + 0.218i)7-s + 0.353·8-s + (−0.429 + 0.744i)9-s + (−0.623 − 1.07i)10-s + (0.832 + 1.44i)11-s + (−0.340 + 0.590i)12-s − 0.277·13-s + (−0.478 + 0.520i)14-s + 2.40·15-s + (−0.125 + 0.216i)16-s + (0.478 + 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.156 - 0.987i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.156 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518725 + 0.442970i\)
\(L(\frac12)\) \(\approx\) \(0.518725 + 0.442970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.58 - 0.578i)T \)
13 \( 1 + T \)
good3 \( 1 + (1.18 + 2.04i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.97 - 3.41i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.76 - 4.78i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.97 - 3.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.18 - 3.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.36 + 2.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.52T + 29T^{2} \)
31 \( 1 + (0.321 + 0.557i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.18 - 2.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.56T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + (-2.97 + 5.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.178 + 0.308i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.12 - 5.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.54 + 6.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.76 - 4.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.10T + 71T^{2} \)
73 \( 1 + (2.80 + 4.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.80 - 4.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.80 - 3.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.88T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.52598146583916273386734673821, −11.92580914927442524517547305758, −11.02191734165429931607652520804, −10.07005018890332772193813309295, −8.310045384568977852214960144323, −7.37946876738990461113464270747, −6.93651324505344213690272177625, −5.90306192213865190281811457548, −4.18125802435354003619689820564, −1.88515135231686826716100185081, 0.821313644474197608080583945920, 3.76037174256457192770360557700, 4.62975791058166607053900413415, 5.43568390486153407678165254783, 7.65796967270142521786461391382, 8.771765165131695769171801267931, 9.274361426659468245705050112859, 10.70273697105785623884847485319, 11.49025426327614124397457529097, 11.84279640584396155092005154478

Graph of the $Z$-function along the critical line